{
  "id": "1010.2536",
  "version": "v2",
  "published": "2010-10-12T23:31:14.000Z",
  "updated": "2011-08-30T04:38:17.000Z",
  "title": "Typicality of normal numbers with respect to the Cantor series expansion",
  "authors": [
    "Bill Mance"
  ],
  "comment": "16 pages",
  "journal": "New York J. Math. 17 (2011), 1-17",
  "categories": [
    "math.NT"
  ],
  "abstract": "Fix a sequence of integers $Q=\\{q_n\\}_{n=1}^\\infty$ such that $q_n$ is greater than or equal to 2 for all $n$. In this paper, we improve upon results by J. Galambos and F. Schweiger showing that almost every (in the sense of Lebesgue measure) real number in $[0,1)$ is $Q$-normal with respect to the $Q$-Cantor series expansion for sequences $Q$ that satisfy a certain condition. We also provide asymptotics describing the number of occurrences of blocks of digits in the $Q$-Cantor series expansion of a typical number. The notion of strong $Q$-normality, that satisfies a similar typicality result, is introduced. Both of these notions are equivalent for the $b$-ary expansion, but strong normality is stronger than normality for the Cantor series expansion. In order to show this, we provide an explicit construction of a sequence $Q$ and a real number that is $Q$-normal, but not strongly $Q$-normal. We use the results in this paper to show that under a mild condition on the sequence $Q$, a set satisfying a weaker notion of normality, studied by A. R\\'enyi in \\cite{Renyi}, will be dense in $[0,1)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-08-30T04:38:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K16",
      "11A63"
    ],
    "keywords": [
      "cantor series expansion",
      "normal numbers",
      "real number",
      "similar typicality result",
      "lebesgue measure"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.2536M"
    }
  }
}